A nonmonotonic horn–like structure of the \(\mathrm {K}^+/\pi ^+\)  ratios was confirmed in various experiments, at AGS, SPS and low RHIC energies [1]}, [2]}, [3]}, [4]}, [5]}. At \(\sqrt{s_{NN}}\lesssim 20~\) GeV, there is a monotonic rapid increase followed by a monotonic rapid decrease. At energies \(\gtrsim 20~\) GeV, the RHIC and LHC energies, the \(\mathrm {K}^+/\pi ^+\)  ratios seem to be nearly energy–independent, i.e., a plateau is formed. Various interpretations have been given in literature [1]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}. All these thermal models which are either based on extensive additive Boltzmann-Gibbs (BG) and/or Tsallis–type nonextensive statistics [14]} aren't succeeded in reproducing the nonmonotonic horn–like structure of the \(\mathrm {K}^+/\pi ^+\)  ratios (long–dashed curve in Fig. REF ).
